Twist spun knots of twist spun knots of classical knots

TL;DR

This paper investigates the properties of iterated twist-spun knots, showing conditions under which they are trivial or non-trivial in higher-dimensional spheres, extending classical knot theory concepts.

Contribution

It provides new results on the triviality and non-triviality of iterated twist-spun knots based on the gcd of twist parameters, advancing understanding of high-dimensional knot constructions.

Findings

m2-twist-spinning of m1-twist-spinning classical knots is trivial if gcd(m1,m2)=1

Provides sufficient conditions for non-triviality of iterated twist-spun knots

Extends classical knot theory to higher dimensions with new criteria

Abstract

A $k$-twist spun knot is an $n+1$-dimensional knot in the $n+3$-dimensional sphere which is obtained from an $n$-dimensional knot in the $n+2$-dimensional sphere by applying an operation called a $k$-twist-spinning. This construction was introduced by Zeeman in 1965. In this paper, we show that the $m_2$-twist-spinning of the $m_1$-twist-spinning of a classical knot is a trivial $3$-knot in $S^5$ if $\gcd(m_1,m_2)=1$. We also give a sufficient condition for the $m_2$-twist-spinning of the $m_1$-twist-spinning of a classical knot to be non-trivial.